Linear equations in two variables
The idea
Some rules connect two changing quantities at once: coins earned and levels completed, cost and weight, distance and time. A linear equation in two variables, like c = 15L + 40, captures such a rule, and a solution is no longer a single number but a pair — a level L together with the coin total c that the rule produces. You already know how to plot pairs on the coordinate plane, and that is where these equations come alive: every solution pair is a point, and together the points line up perfectly.
The straight line through those points IS the solution set, drawn rather than listed. That resolves the big misconception that an equation must have one answer: a two-variable linear equation has infinitely many solution pairs, and the line shows them all at once. To test whether a pair belongs, substitute both numbers and see whether the two sides agree; to find a missing partner, plug in the value you know and solve the leftover one-variable equation.
Worked example
A game gives you 40 coins to start and 15 coins per level, so your total is c = 15L + 40 after L levels. How many coins do you have after level 6, and at which level do you first reach exactly 190 coins?
- For the first question L is known, so substitute L = 6: c = 15 × 6 + 40 = 90 + 40 = 130 coins.
- For the second question c is known, so set 15L + 40 = 190 and subtract 40 from both sides: 15L = 150.
- Divide both sides by 15: L = 150 ÷ 15 = 10, so level 10 brings the total to exactly 190.
- Read both results as points: (6, 130) and (10, 190) sit on the same line, two of the infinitely many pairs this rule makes true.
Answer. You have 130 coins after level 6, and you reach exactly 190 coins at level 10.
Check your understanding
- Why does a two-variable equation have infinitely many solutions while a one-variable equation usually has just one?
- How would the line and its meaning change if the game gave no starting coins at all?
- What does it mean graphically when a pair you test makes the two sides of the equation disagree?
- Given any one solution pair, how could you generate several more without solving from scratch?
Build the foundations first
Linear equations in two variables builds on these concepts. If any feel shaky, start there.